On infinite multiplicative Sidon sets

Abstract

We prove that if $A$ is an infinite multiplicative Sidon set, then $\underset{n\to \infty }{\mathrm{liminf}}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n{)}^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\underset{n\to \infty }{\mathrm{liminf}}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n{)}^3}}>0$.